Perpendicular wires: B field

Okay, this is a conceptual problem that I'm sure is easy, but is bugging me:

I have two wires lying in the xy-plane. They're perpendicular, with one wire lying along the y-axis and on along the x-axis. The current in each wire points along the y and x axes, respectively.

Now: What's the magnetic field like in the plane?

From the right hand rule, each wire has a field perpendicular to it and "wrapping" around it. So, does the field from the two wires cancel? it seems to me that it would, because in the first quadrant (for instance) the field from the y-axis wire points into the page, while the field for the x-axis wire points out of the page. Is this right?

The magnetic field will cancel for some specific points (actually lines).
If the two currents are equal, the field will be zero along either the line x=y or the line x=-y.
In rest, the fields will have different magnitudes and the net field will be non-zero.